Sigmoid Function

Question

Typically sigmoid function is calculated as

    1/(1 + exp(-x))

I see sometimes it is calculated as

    1 - 1/(1 + exp(x))

or even

   exp(x)/(1 + exp(x))

Could you clarify the difference please?

Answer 1

$$ \frac {1}{1+\exp(-x)} = \frac {1}{1+e^{-x}} = \frac {1}{1+e^{-x}} \cdot \frac {e^x}{e^x} = \frac {e^x}{e^x+1} = \frac {e^x}{1+e^x} = \frac {\exp(x)}{1+\exp{x}} $$ and also $$ 1-\frac {1}{1+\exp(x)} = 1-\frac {1}{1+e^{x}} = \frac{1+e^x}{1+e^x} - \frac{1}{1+e^x} = \frac{(1+e^x)-1}{1+e^x} = \frac {e^x}{1+e^x} = \frac {\exp(x)}{1+\exp{x}} $$